Atomic cross sections

Cross sections are approximated with semi-analytic expressions, obtained from E.Havlickova but of unknown origin. For the purposes of calculating these cross-sections, any temperatures below 1eV are set to 1eV. The charge exchange cross-section is approximated as:

\[\begin{split}\sigma_{iz} = \left\{\begin{array}{ll} 10^{-14} T^{1/3} & \textrm{if $T \ge 1$eV} \\ 10^{-14} & \textrm{if $T < 1$eV} \end{array}\right.\end{split}\]

with units of \([\textrm{m}^3/\textrm{s}]\). Ionisation is calculated as

\[\begin{split}\sigma_{cx} = \left\{\begin{array}{ll} 5.875\times 10^{-12}\cdot T^{-0.5151} \cdot 10^{-2.563/\log_{10}T} & \textrm{if $T \ge 20$eV} \\ 10^{-6}\cdot T^{-3.054}\cdot 10^{-15.72\exp\left(-\log_{10}T\right) + 1.603\exp\left(-\log^2_{10}T\right)} & \textrm{if $1$eV $ < T < 20$eV} \\ 7.638\times 10^{-21} & \textrm{if $T \le 1$eV}\end{array}\right.\end{split}\]

Recombination rates are calculated using a \(9\times 9\) table of coefficients so is not reproduced here.

Cross-sections [Thanks to E.Havlickova and H.Willett]

Fig. 1 Cross-sections [Thanks to E.Havlickova and H.Willett]

Plots of these cross-sections are shown in figure 1. There are a few anomalies with this: charge exchange always has the highest cross-section of any process, and ionisation has a jump at \(20\)eV. The ionisation and charge exchange rates do not depend on density, but recombination does so a typical range of values is shown.